Korean J. Math.  Vol 26, No 3 (2018)  pp.373-385
DOI: https://doi.org/10.11568/kjm.2018.26.3.373

Fuzzy homomorphism theorems on groups

Gezahagne Mulat Addis


In this paper we introduce the notion of a fuzzy kernel of a fuzzy homomorphism on groups and we show that it is a fuzzy normal subgroup of the domain group. Conversely, we also prove that any fuzzy normal subgroup is a fuzzy kernel of some fuzzy epimorphism, namely the canonical fuzzy epimorphism. Finally, we formulate and prove the fuzzy version of the fundamental theorem of homomorphism and those isomorphism theorems.


Fuzzy mappings; fuzzy homomorphisms; fuzzy normal subgroups; quotient groups induced by fuzzy normal subgroups; fuzzy isomorphism theorems

Subject classification

03E72, 08A72, 20A99


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N. Ajmal, Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient group, Fuzzy Sets and Systems 61 (1994), 329–339. (Google Scholar)

A. B. Chakraborty and S. S. Khare, Fuzzy homomorphism and algebraic struc- tures, Fuzzy Sets and Systems 59 (1993) 211–221. (Google Scholar)

F. P. Choudhury, A. B. Chakraborty and S. S. Khare, A Note on Fuzzy Subgroups and Fuzzy Homomorphism, J. Math. Anal. Appl 131 (1988), 537–553. (Google Scholar)

T. W. Hungerford, Algebra: A Graduate Text in Mathematics, Springer, New York (1980). (Google Scholar)

Y. L. Liu, Quotient groups induced by fuzzy subgroups, Quasigroups and Related Systems 11 (2004), 71–78. (Google Scholar)

Y. L. Liu, Quotient rings induced via fuzzy ideals, J. Computations and Appl. Math. 8 (2001), 631–643. (Google Scholar)

D. S. Malik and John N. Mordeson, Fuzzy homomorphism of rings, Fuzzy sets and systems 46 (1992) 139–146. (Google Scholar)

N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inf. Sci. 34 (1984), 225–239. (Google Scholar)

V. Murali, A study of universal algebras in fuzzy set theory, Doctoral Thesis, Department of Mathematics, Rhodes University (1987). (Google Scholar)

V. Murali, Fuzzy congruence relation, Fuzzy Sets and Systems 41 (1991), 359– 369. (Google Scholar)

A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl. 35 (1971), 512–517. (Google Scholar)

M. A. Samhan and T. M. G. Ahsanullah, Fuzzy Congruences on Groups and Rings, Internat. J. Math. Math. Sci. 17 (1994), 469–474. (Google Scholar)

B. K. Sarma and T. Ali, Weak and Strong Fuzzy Homomorphisms of Groups, J. Fuzzy Math. 12 (2004), 357–368. (Google Scholar)

L. A. ZADEH, Fuzzy Sets, Information and Control 8 (1965), 338–353. (Google Scholar)

X. Zhou, D. Xiang and J. Zhan Quotient rings via fuzzy congruence relations, Italian Journal of Pure and Applied Mathematics 33 (2014), 411–424. (Google Scholar)


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